Space of modular forms of level 6223, weight 2, and trivial character

• Label: 6223.2.a
• Level: $6223$
• Weight: $2$
• Character orbit: 6223.a
• Rep. character: $\chi_{6223}(1,\cdot)$
• Character field: $\Q$
• Dimension: $431$
• Newform subspaces: $20$
• Sturm bound: $1194$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(1194\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6223))\).

	Total	New	Old
Modular forms	604	431	173
Cusp forms	589	431	158
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)	\(127\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(96\)
\(+\)	\(-\)	$-$	\(116\)
\(-\)	\(+\)	$-$	\(117\)
\(-\)	\(-\)	$+$	\(102\)
Plus space		\(+\)	\(198\)
Minus space		\(-\)	\(233\)

Trace form

\( 431 q + 2 q^{2} + 4 q^{3} + 436 q^{4} + 4 q^{5} + 6 q^{6} + 437 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6223))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	127
6223.2.a.a	$6223$	$2$	6223.a	1.a	$1$	$1$	$1$	$49.691$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{5}-3q^{8}-3q^{9}-2q^{10}+\cdots\)
6223.2.a.b	$6223$	$2$	6223.a	1.a	$1$	$1$	$1$	$49.691$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{5}-3q^{8}-3q^{9}+2q^{10}+\cdots\)
6223.2.a.c	$6223$	$2$	6223.a	1.a	$1$	$2$	$2$	$49.691$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(3\)	\(-4\)	\(3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-2q^{3}+3\beta q^{4}+(2-\beta )q^{5}+\cdots\)
6223.2.a.d	$6223$	$2$	6223.a	1.a	$1$	$2$	$2$	$49.691$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(4\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+2q^{3}+3\beta q^{4}+(-2+\beta )q^{5}+\cdots\)
6223.2.a.e	$6223$	$2$	6223.a	1.a	$1$	$3$	$3$	$49.691$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(-3\)	\(3\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots\)
6223.2.a.f	$6223$	$2$	6223.a	1.a	$1$	$4$	$4$	$49.691$	4.4.3981.1	None	✓	$1$	$0$	\(2\)	\(-1\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{3})q^{2}-\beta _{1}q^{3}+(1-\beta _{2})q^{4}+\cdots\)
6223.2.a.g	$6223$	$2$	6223.a	1.a	$1$	$4$	$4$	$49.691$	4.4.3981.1	None	✓	$1$	$1$	\(2\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}-\beta _{3})q^{2}+\beta _{1}q^{3}+(1-\beta _{2})q^{4}+\cdots\)
6223.2.a.h	$6223$	$2$	6223.a	1.a	$1$	$7$	$7$	$49.691$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(-3\)	\(-8\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{2}-\beta _{3}-\beta _{5}-\beta _{6})q^{3}+\cdots\)
6223.2.a.i	$6223$	$2$	6223.a	1.a	$1$	$12$	$12$	$49.691$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-7\)	\(4\)	\(7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+\beta _{10}q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
6223.2.a.j	$6223$	$2$	6223.a	1.a	$1$	$15$	$15$	$49.691$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{9}q^{3}+(1+\beta _{2})q^{4}-\beta _{8}q^{5}+\cdots\)
6223.2.a.k	$6223$	$2$	6223.a	1.a	$1$	$16$	$16$	$49.691$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(4\)	\(9\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{10}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
6223.2.a.l	$6223$	$2$	6223.a	1.a	$1$	$20$	$20$	$49.691$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{9}q^{3}+(1+\beta _{2})q^{4}+\beta _{16}q^{5}+\cdots\)
6223.2.a.m	$6223$	$2$	6223.a	1.a	$1$	$26$	$26$	$49.691$		None	✓	$2$	$1$	\(-8\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
6223.2.a.n	$6223$	$2$	6223.a	1.a	$1$	$34$	$34$	$49.691$		None	✓	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
6223.2.a.o	$6223$	$2$	6223.a	1.a	$1$	$38$	$38$	$49.691$		None	✓	$1$	$1$	\(-2\)	\(-11\)	\(-16\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
6223.2.a.p	$6223$	$2$	6223.a	1.a	$1$	$38$	$38$	$49.691$		None	✓	$1$	$0$	\(-2\)	\(11\)	\(16\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
6223.2.a.q	$6223$	$2$	6223.a	1.a	$1$	$40$	$40$	$49.691$		None	✓	$1$	$1$	\(-3\)	\(-6\)	\(-24\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
6223.2.a.r	$6223$	$2$	6223.a	1.a	$1$	$40$	$40$	$49.691$		None	✓	$1$	$0$	\(-3\)	\(6\)	\(24\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
6223.2.a.s	$6223$	$2$	6223.a	1.a	$1$	$54$	$54$	$49.691$		None	✓	$2$	$1$	\(-14\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
6223.2.a.t	$6223$	$2$	6223.a	1.a	$1$	$74$	$74$	$49.691$		None	✓	$2$	$0$	\(18\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6223))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6223)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(127))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(889))\)\(^{\oplus 2}\)